We plot in Fig. REF (a) the dynamic evolutions of \(P_{b,1}\)  and \(P_{a,2}\)  by numerically calculating Eq. (REF ). The separation distance \(d\)  between adjacent coupling ports is well tailored such that \(2\phi =\pi \) . According to Ref. [1]}, [2]}, the emitters become decoherence-free in this case yet can still interact with each other in the absence of the direct coupling. However, as show in Fig. REF (a), the excitation transfer becomes reciprocal in this case, although the coupling phase \(\theta =\pi /2\)  is nontrivial (we have also checked the dynamic evolutions with other values of \(\theta \) , which always show reciprocal transfer). Moreover, we can find that in the presence of considerable time delay, the excitation transferred between two emitters still shows a damped oscillation although the damping is much weaker than that in Fig. REF (b). This is because the emitters are not exactly decoherence-free due to the retarded self interference effects. As \(\eta \)  (i.e., \(d\) ) decreases gradually, the non-Markovian retarded effect becomes more and more negligible such that the emitters tends to be completely decoherence-free, as shown in Fig. REF (b).
